Theorem com3ii 439

Description: Lemma 3(ii) of Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
comcom.1	a C b

Assertion

Ref	Expression
com3ii	(a ^ (a_|_ v b)) = (a ^ b)

Proof of Theorem com3ii

Step	Hyp	Ref	Expression
1		comcom.1	. . . . . 6 a C b
2	1	comcom 435	. . . . 5 b C a
3	2	comd 438	. . . 4 b = ((b v a) ^ (b v a_|_))
4	3	lan 70	. . 3 (a ^ b) = (a ^ ((b v a) ^ (b v a_|_)))
5		anass 69	. . . . 5 ((a ^ (b v a)) ^ (b v a_|_)) = (a ^ ((b v a) ^ (b v a_|_)))
6	5	ax-r1 34	. . . 4 (a ^ ((b v a) ^ (b v a_|_))) = ((a ^ (b v a)) ^ (b v a_|_))
7		ax-a2 30	. . . . . . 7 (b v a) = (a v b)
8	7	lan 70	. . . . . 6 (a ^ (b v a)) = (a ^ (a v b))
9		a5c 113	. . . . . 6 (a ^ (a v b)) = a
10	8, 9	ax-r2 35	. . . . 5 (a ^ (b v a)) = a
11		ax-a2 30	. . . . 5 (b v a_|_) = (a_|_ v b)
12	10, 11	2an 72	. . . 4 ((a ^ (b v a)) ^ (b v a_|_)) = (a ^ (a_|_ v b))
13	6, 12	ax-r2 35	. . 3 (a ^ ((b v a) ^ (b v a_|_))) = (a ^ (a_|_ v b))
14	4, 13	ax-r2 35	. 2 (a ^ b) = (a ^ (a_|_ v b))
15	14	ax-r1 34	1 (a ^ (a_|_ v b)) = (a ^ b)

Syntax hints:   = wb 1   C wc 3  _|_wn 4   v wo 6   ^ wa 7

This theorem is referenced by:  fh1 451  fh2 452

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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